Higher Derivations in Invariant Theory
William Traves
20 March 2003
Much of the wonderful invariant theory of the 19th century worked over
fields of characteristic zero, while the theory for prime characteristic lagged
behind. However, the Frobenius (pth power) map in characteristic p > 0
leads to a rich theory of invariants in prime characteristic. This theory is
closely bound up with the Steenrod Algebra, which allows us to derive new
invariants from known invariants.
I won’t go into the history of the Steenrod Algebra except to say that
it was developed to study certain cohomology rings in Algebraic Topology.
We’ll take a more elementary (and algebraic) approach to the Steenrod Algebra which has been popularized by L. Smith and R. Wood.
1
The Steenrod Algebra
Notation: Let R = k[x1 , . . . , xn ] be a polynomial ring over a field k = Fq of
characteristic p > 0 (hence q = pe ). We think of R as the ring of polynomial
functions on the affine space k n .
Define a map φ : R → R[[t]] on each xi via
xi �→ xi + xqi t
and extend φ to a ring homomorphism. For example,
xi xj �→ (xi xj ) + (xqi xj + xi xqj )t + (xqi xqj )t2 .
Define operators Qi : R → R to be the k-linear maps defined by applying
φ and then extracting the coefficient of ti from the resulting polynomial. For
instance,
Q1 (xi xj ) = xqi xj + xi xqj .
1
Definition: The operators Qi form a k-subalgebra of the k-linear endomorphisms of R called the Steenrod Algebra and denoted A.
The operators Qi satisfy a product rule that is an extension of the rule
we teach in Calculus:
�
Theorem 1.1 (Cartan Formula). Qk (f g) = i+j=k Qi (f )Qj (g).
Example 1.2. Q1 (xi xj ) = Q1 (xi )Q0 (xj ) + Q0 (xi )Q1 (xj ) = xqi xj + xi xqj .
As well, the operators {Qi : i > 0} are unstable in the sense that if fd
is a homogeneous polynomial of degree d then Qi (fd ) is fdq if i = d and is 0
if i > d. In particular, none of the operators are zero (though they are all
nilpotent!). [check]
Now Gl(n, Fq ) acts on R by linear change of variables. Since the coefficients of the matrices are in Fq , they satisfy aq = a. This implies that the
action of Gl(n, Fq ) on the vector space with basis {x1 , . . . , xn } is the same
as its action on the vector space with basis {xq1 , . . . , xqn } (the group action is
represented by the same matrix). As a consequence, the Steenrod operations
commute with linear group actions. If G is any group that acts linearly on
R, then the Steenrod operations Qi induce maps RG → RG : if r ∈ RG and
g ∈ G, then g · Qi (r) = Qi (g · r) = Qi (r). Here RG is the ring of invariants of
the group action and it should be thought of as the ring of functions on the
quotient space k n /G.
The Qi raise degree and preserve invariants. So they can be used to create
new (higher degree) invariants from known invariants. These operators have
another important property: for each operator Qi , there is a power pe so
e
that Qi is Rp -linear. This follows from Cartan’s Formula and the unstability
property: for pe > i we have
�
e
e
Qi (rp f ) =
Qm (rp )Qn (f )
m+n=i
e
= Q0 (rp )Qi (f )
e
= rp Qi (f )
Much is known about the structure of the Steenrod Algebra. The complete set of relations, called the Adem relations, has been determined. These
can be derived in several beautiful ways and are encoded by the BulletMacdonald identity. Much is also known about the structure of RG as a
module over A; for instance, there are only finitely many stable prime ideals
in RG and these are generated by intervals in the Dixon invariants.
The Steenrod Algebra can also be interpreted as a ring of differential
operators. We take up this theme in the next section.
2
2
Differential Operators
Grothendieck defined a ring of differential operators on algebraic variety. His
definition is abstract and quite complicated, but it can be easily understood
for k n . The ring of differential operators on Cn (or equivalently, on R =
C[x1 , . . . , xn ]) is D(R) = C[x1 , . . . , xn , ∂1 , . . . , ∂n ]. This is sometimes called
the Weyl algebra. It is not quite a polynomial ring: the generators satisfy
the relations
[∂i , xj ] = ∂i xj − xj ∂i = δij .
(This is just the ordinary product rule: ∂i = ∂x∂ i .)
When k has prime characteristic p > 0, something odd happens:
∂i (xpi ) = pxp−1
= 0.
i
Moreover, ∂ip ≡ 0. To get around this problem, we introduce the divided
1 ∂m
powers operators, ∂im = m!
. We have ∂im (xti ) = i(i−1)···(i−m+1)
xt−m =
∂xm
m!
i
� i � t−m
x . Then
m
D(k[x1 , . . . , xn ] = k[xi , ∂im ]m>0 .
We will want to discuss the relation between D(R) and D(RG ) so we need
to define the ring of differential operators on affine varieties. There are two
equivalent ways to do this. If we think of RG as S/I then
D(RG ) = D(S/I) =
{θ ∈ D(S) : θ(I) ⊆ I}
.
ID(S)
Alternatively, we could think of RG as a subalgebra of R. From this point of
view
D(RG ) = {θ ∈ D(R) : θ(RG ) ⊆ RG }/{θ ∈ D(R) : θ(RG ) = 0}.
There is a third description of the ring of differential operators that makes
it clear that the Qi are differential operators.
Theorem 2.1 (K. E. Smith). In characteristic p > 0, the ring of differential
operators on R is just the subalgebra of all endomorphisms of R that are
e
Rp -linear for some power pe .
3
e
Since A is generated by Rp -linear maps, the Steenrod Algebra is a subalgebra of the ring of differential operators. In fact, we can write
�
Qi =
xqa ∂ a .
a
If we filter the ring of differential operators by the order of the operators
appearing, then Qi is a differential operator of order i.
If G is a group that acts on R, then it can be made to act on differential
operators too. For g ∈ G and DinD(R), set gD to be the operator such that
(gD)(r) = g · D(g −1 · r). Note that if g · D = D then g commutes with D
(and hence D defines an operator on RG ).
Rings of differential operators find application in a wide range of mathematical fields. For instance, they can be used to model quantum mechanics,
they are used to study symplectic manifolds, and they play an increasingly
important role in commutative algebra (through a mysterious relationship
with tight closure).
Questions: When is the ring of differential operators D(RG ) finitely generated, left or right Noetherian? When is D(RG ) a simple ring? And when
is RG a simple module over D(RG )? What about the same questions for
GrD(RG )? How do the algebras D(R)G andD(RG ) relate?
Kantor and Levasseur studied the case when G is a finite group. In
this case D(RG ) is finitely generated, simple and left and right Noetherian.
Moreover the map π : D(R)G → D(RG ) is a surjection whenever G contains
no pseudoreflections.
Schwarz, Levasseur, Van den Bergh, Musson, Stafford and many others
have studied the classical groups acting on a polynomial ring in characteristic
zero. Here, D(RG ) (and even GrD(RG )) are known to be finitely generated
when G is commutative. As well, the map D(R)G → D(RG ) is a surjection.
However, there are representations of Sl2 (C) for which the map D(R)G →
D(RG ) is not a surjection (Schwarz).
The question of simplicity is quite interesting. If D(S) is simple, then S
is a simple module over D(S). [Reason: AnnD(S) (S/I) is a nonzero two-sided
ideal of D(S) and hence must be all of D(S) – i.e. I = S.]
There is a general feeling (perhaps misguided) that the characteristic zero
case is better than the prime characteristic case. However, when R has prime
characteristic, D(RG ) is always simple. [due to Van den Bergh and K. E.
Smith] This remains an open problem in general in characteristic zero.
4
3
Higher Derivations
The Steenrod operations are an example of a higher derivation, collections of
operators that generalize the behaviour of derivations on commutative rings.
Let R = k[x1 , . . . , xn ]/I be a finitely generated ring. A higher derivation from R to itself is an infinite collection of maps of k-algebras {D0 =
idR , D1 , D2 , . . .} from R to R that patch together using the product rule:
�
Dm (f g) =
Di (f )Dj (g).
i+j=m
Example 3.1. The Steenrod operators {Q0 , Q1 , . . .} determine a higher derivation from R = k[x1 , . . . , xn ] to itself.
Example 3.2. In characteristic zero, any derivation d on R determines a
higher derivation
1
Dk = dk .
k!
d
For instance, the derivation dx on k[x] induces a higher derivation on the
polynomial ring.
Each higher derivation from R to R gives rise to a map of k-algebras
φ : R → R[[t]]
r �→ D0 (r) + D1 (r)t + D2 (r)t2 + · · ·
analogous to the case of the Steenrod operators.
There is no unstability result for the higher derivations in general, but
we do have the following nice fact in prime characteristic p > 0:
�
e
(Dm/pe (r))p if pe |m
pe
Dm (r ) =
0
otherwise
e
Proof: Apply the ring homomorphism φ to rp :
e
e
e
rp + D1 (rp )t + D2 (rp )t2 + · · ·
e
= φ(rp )
e
e
= (φ(r))p = (r + D1 (r)t + D2 (r)t2 + · · · )p
e
e
e
e
e
= rp + D1 (r)p tp + D2 (r)p t2p + · · ·
Theorem 3.3. Each higher derivation is a collection of differential operators.
5
Proof: The previous result, combined with the product rule, implies that
e
each Di is Rp -linear for some power pe .
The algebra of higher derivations a ring S is just the S-algebra HDer(S)
generated by the components of all higher derivations on S. We have
RG A ⊆ HDer(RG ) ⊆ D(RG ).
Here, RA is the RG -algebra generated by the Steenrod operators.
It makes sense to ask when these inclusions become equalities. When RG
is regular (eg. a polynomial algebra), RG A �= HDer(RG ) but HDer(RG ) =
D(RG ). The first statement follows from the second since in this case D(RG )
is a simple ring so RG is a simple module over HDer(RG ) = D(RG ) but RG
contains a nontrivial RA-stable ideal generated by the Dixon invariants. The
fact that HDer(RG ) = D(RG ) when RG is regular follows from a delicate
argument: first prove the result for polynomial rings and then lift the result
to Spec(RG ), which is an étale cover of an affine space.
Conjecture 3.4 (Nakai’s Conjecture). HDer(S) = D(S) if and only if S is
regular.
The conjecture was proven for S = RG a ring of invariants on a polynomial ring under a finite group action (Ishibashi). This is easy to see in
the prime characteristic case where RG is not regular because D(RG ) is a
simple algebra but HDer(RG ) is not: the singular locus of R corresponds to
a radical ideal that is HDer-stable. In this case, I do not know when RA
can equal HDer(RG ).
4
Jet Spaces
Higher derivations are supposed to extend our notion of derivations (maps
satisfying the usual product rule d(f g) = f d(g) + d(f )g). The derivations
are represented by the module ΩR/k of k-differentials on R. That is, there
is a map d : R → ΩR/k such that whenever D : R → R is a derivation,
there exists a unique R-module homomorphism φ : ΩR/k → R such that
φ ◦ d = D. We say that ΩR/k represents the derivations since each derivation
is determined by φ ∈ HomR (ΩR/k , R).
There is a similar construction that produces a k-algebra HSR/k representing the higher derivations. Analogous to the derivation case, there is a
6
sequence of maps (d0 , d1 , . . .) from R to HSR/k such that each higher derivation {D0 , D1 , . . .} from R to R determines a unique map φ : HSR/k → R of
k-algebras by φ ◦ di = Di .
4.1
Higher Derivations to k
We can also consider higher derivations from R to k. The whole theory
goes through as before: higher derivations
are collections of k-algebra maps
�
Di : R → k with Dk (f g) =
i+j=k Di (f )Dj (g). These are once again
determined uniquely by a k-algebra map φ : HSR/k → k.
So we see that the collection of higher derivations HDerk (R, k) is isomorphic to Homk−alg (HSR/k , k), the collection of k-algebra maps from HSR/k to
k. (*)
On the other hand, each higher derivation {D0 , D1 , . . .} from R to k
determines a map of k-algebras φ : R → k[[t]] given by:
φ(r) = D0 (r) + D1 (r)t + D2 (r)t2 + · · ·
This ring homomorphism is determined by the images φ(xi ) of the variables
xi . We require that f (φ(x1 ), . . . , φ(xn )) = 0 for all polynomials f in the
defining ideal I.
Thus, HDerk (R, k) is isomorphic to Homk−alg (R, k[[t]]). (**)
From (*) and (**) we get that Homk−alg (R, k[[t]]) ∼
= Homk−alg (HSR/k , k).
Taking Spec’s we see that
[maps Spec(k[[t]]) → Spec(R)] ∼
= [maps Spec(k) → SpecHSR/k ] ∼
= SpecHSR/k .
Thus, the jet space J(Spec(R)) := Spec(HSR/k ) parameterizes arcs on Spec(R).
We have a natural map π : J(Spec(R)) → Spec(R) that sends an arc to the
closed point that it passes through. We can understand this map using the
ring map. Each arc γ gives rise to a map R → k[[t]] (here we are using
the antiequivalence of categories between schemes and rings). This assigns
a power series ai + HOT (t) to each coordinate function xi . Just looking at
the constant terms of these power series gives a point π(γ) = (a1 , . . . , an ) in
k n that lies on X = Spec(R). We call the preimage of a point O under this
map π, the set of arcs through O (it is actually a scheme).
7
4.2
Nash’s Conjecture
Nash conjectured a relationship between the jet space and resolution of singularities. To state the conjecture, suppose that X is an algebraic surface
with an isolated singular point at the origin O. It is a classical result that we
can resolve the singularities of X by repeatedly blowing up points (which can
introduce 1-dimensional singularities) and then normalizing (which replaces
the 1-dimensional singularities by singularities at points). This leads to a
smooth surface X̄ called the minimal model of X. It is minimal in the sense
that if any other variety Z maps to X and is birational, then the map must
factor through γ : X̄ → X.
The essential exceptional divisors of X are just the preimage γ −1 (O) of
the isolated singularity. This is a union of curves ∪Ei on the minimal model
X̄. Nash produced an injective map from the components of the jet space of
arcs on X through O to the set of exceptional divisors. He conjectured that
this map is surjective. This conjecture is true for many classes of varieties
(eg. toric varieties, surfaces, etc), however Kollar and Ishii recently showed
that the conjecture is false in general.
4.3
Motivic Integration
The jet space also plays a crucial role in Kontsevich’s theory of motivic
integration. He developed this theory to prove a conjecture of Batyrev: two
birationally equivalent Calabi-Yau manifolds have the same Hodge numbers.
For the purposes of this overview, it is not really important to know what
Calabi-Yau manifolds are, or to know what Hodge numbers are. All we need
to know is that the Hodge numbers (hp,q = dimH p (Ωq , X)) are numerical
invariants
of X. We have a map that sends each Calabi-Yau variety X to
�
p q
h
u
v ∈ Z[u, v].
p,q p,q
Kontsevich showed that this map factors through another map X → M,
that sends the variety X to an element in a ring that he calls the motivic
ring. The ring M consists of formal Z-linear combinations of subvarieties in
X enlarged by adding inverses of some of the elements; addition corresponds
to disjoint union of varieties while multiplication corresponds to direct product of varieties. Kontsevich interprets this map to the motivic ring as an
integration map on the arc space of X. Roughly, Kontsevich develops a new
theory of integration in which the space we integrate over is the arc space of
X, the measurable sets are certain subsets of J(X) called cylinder sets, the
8
integrals take values in this odd motivic ring and we have a nice change of
variables formula for integration. He uses this change of variables formula to
show that when X and Y are birational Calabi-Yau manifolds their images
in the motivic ring are equal. Thus, so are their images in the ring Z[u, v].
Motivic integration also finds many other interesting applications, including zeta functions, p-adic integration, string theory and mirror symmetry.
There is a nice seminar at Berkeley this year that is investigating applications of motivic integration to characterizations of singularities.
9